Analysis of congestion periods of an m/m/∞-queue

Abstract

c-congestion period of an m/m/ ∞-queue is a period during which the number of customers in the system is continuously above level c. Interesting quantities related to a c-congestion period are, besides its duration Dc, the total area Ac above c, and the number of arrived customers Nc. In the literature Laplace transforms for these quantities have been derived, as well as explicit formulae for their means. Explicit expressions for higher moments and covariances (between Dc, Nc and Ac), however, have not been found so far. This paper presents recursive relations through which all moments and covariances can be obtained. Up to a starting condition, we explicitly solve these equations; for instance, we write E Dc2 explicitly in terms of E D02. We then find formulae for these starting conditions (which directly relate to the busy period in the m/m/ ∞ queue). Finally, a c-intercongestion period is defined as the period during which the number of customers is continuously below level c. Also for this situation a recursive scheme allows us to explicitly compute higher moments and covariances. Additionally we present the Laplace transform of a so-called intercongestion triple of the three performance quantities. It is also shown that expressions for the quantities of a c-intercongestion period can be used in an approximation for the c-congestion period. This is especially useful as the expressions for the c-intercongestion period are numerically more stable than those for the c-congestion period. © 2007 Elsevier Ltd. All rights reserved.